In this work, we study the Cauchy problem of the sixth-order Boussinesq equation. We shall use the $[k; Z]$-multiplier norm method to get a bilinear estimate on nonlinear term, and the local well-posedness on $H^s(\mathbb{R})$ is obtained for $s\gt-\frac{3}{4}$.

This note introduces bilinear estimates intended as a step towards an $L^\infty$-endpoint Kato-Ponce inequality. In particular, a bilinear version of the classical Gagliardo-Nirenberg interpolation inequalities for a product of functions is proved.

We discuss the existence of the blow-up solution for multi-component parabolic-elliptic drift-diffusion model in two space dimensions. We show that the local existence, uniqueness and well posedness of a solution in the weighted $L^2$ spaces. Moreover, we prove that if the initial data satisfies a threshold condition, the corresponding solution blows up in a finite time. This is a system case for the blow up result of the chemotactic and drift-diffusion equation proved by Nagai [22] and Nagai-Senba-Suzuki [24] and gravitational interaction of particles by Biler-Nadzieja [4], [5]. We generalize the result in Kurokiba-Ogawa [17] for multi-component problem and give a sufficient condition for the finite time blow up of the solution.

We consider the general semilinear parabolic equation $u_t-\Delta u=\int_0^t k(t,s)|u|^{p-1}u(s)ds$ in $(0,T)\times \mathbb{R}^N$ with $p > 1$. The function $k:\{(t,s); s < t\} \to \mathbb{R}$ is continuous and verifies a scaling property. We prove the existence and non existence results for initial data in the space $L^r(\mathbb{R}^N)$ with $1\leq r < \infty$.

We study the existence and uniqueness of the bounded solutions to a fractional nonlinear porous medium equation with a variable density in a one space dimension.

We discuss stable solutions of nonlinear elliptic equations on the whole space.

In this paper, we study the long time behavior in the distribution sense of solutions for a stochastic Klein-Gordon-Schrödinger type system, which is defined in a unbounded domain. The existence of one stationary measure from any moment-finite initial data in the space $H^{1}(\mathbb{R})\times H^{1}(\mathbb{R})\times L^{2}(\mathbb{R})$ is proven and then a global measure attractor is constructed consisting of probability measures supported on $H^{2}(\mathbb{R})\times H^{2}(\mathbb{R})\times H^{1}(\mathbb{R}). $

We present a result on existence of stationary solutions of a system coupling singular Navier-Stokes equations to a enthalpy-heat equation. This system may model the solidification process of certain classes of materials by taking into consideration the possibility of flow in the melt; thus, the singular Navier-Stokes equations only holds in the a priori unknown molten region and one has a free-boundary value problem to be solved. To obtain solutions for such system, we initially consider a sequence of approximate problems associated to an appropriate regularizations of the original one, which are obtained by a suitable modification such that the approximate equations for the flow hold in the entire domain. After analyzing these approximate problems, by using compactness arguments, we take limits to obtain generalized solutions of the original problem.

In this article we are interested in energy estimates for hyperbolic initial boundary value problem when surface waves occur. More precisely, we construct rigorous geometric optics expansions for so-called elliptic and mixed frequencies and we show, using those expansions, that the amplification phenomenon is greater in the case of mixed frequencies. As a consequence, this result allow us to give a partial classification of weakly well-posed hyperbolic initial boundary value problems according to the region where the uniform Kreiss Lopatinskii condition degenerates.

We show the existence of at least three different continuations beyond blow-up for a backward self-similar solution of a supercritical Fujita equation. One of these extended solutions cannot be approximated by classical entire solutions in a specific way given by the scaling invariance of the equation, while the minimal continuation is known to be accessible by a family of such entire solutions.

We prove that extremals for second order Rellich-Sobolev inequalities have a constant sign. Then, we show that the optimal constants in Rellich-Sobolev inequalities on a bounded domain $\Omega$ and under Navier boundary conditions do not depend on $\Omega$.